Alternative algebra

From Academic Kids

In abstract algebra, an algebra is called alternative if (xx)y=x(xy) and y(xx)=(yx)x for all x and y in the algebra, that is, if the multiplication is alternative.

Equivalently, an algebra is alternative if and only if the subalgebra generated by any two of its elements is associative. The equivalence of the two definitions is known as Artin's Theorem.

For any two elements x and y in an alternative algebra another simple identity holds: (xy)x = x(yx). This is called the flexible law.

Every associative algebra is obviously alternative, but so too are some non-associative algebras such as the octonions.

Alternativity in algebras is a condition inbetween associativity and power associativity.

Template:Math-stubde:Alternativkörper fr:alternativité

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